Voice communication can still be used for anything out of the ordinary despite automating the common case.

Almost all voice transmissions are routine instructions/clearances from ground to air, with the pilots reading them back to reduce the chance of errors. In fact, this already exists and is in wide use in (at least) the US, EU, and in transoceanic airspace.

Of course, now you have two systems that can fail, and reducing reliance on the older one can easily cause automation complacency (which is a well-researched source of errors) and require more frequent refresher courses if the skill is not practiced on a continuos basis.

I suspect that that these are the reasons it's not commonly used for approach and tower operations: There's a lot more spontaneous and/or nonstandard stuff happening in those flight phases, and as you say you don't want a pilot's eyes on a tiny screen/keyboard instead of on their instruments or out the window.